| curveId | Category | s1 | s2 | s3 |
|---|---|---|---|---|
| 1 | A | -0.21 | 0.21 | -0.02 |
| 2 | A | -0.20 | -0.07 | 0.02 |
| 3 | A | -0.15 | 0.16 | 0.00 |
| 51 | B | 0.21 | 0.04 | 0.05 |
| 52 | B | 0.24 | 0.14 | -0.03 |
| 53 | B | 0.27 | 0.17 | 0.06 |
Functional PCA
July 1, 2024
funData, MFPCAfdalandmarkregUtils| curveId | Category | s1 | s2 | s3 |
|---|---|---|---|---|
| 1 | A | -0.21 | 0.21 | -0.02 |
| 2 | A | -0.20 | -0.07 | 0.02 |
| 3 | A | -0.15 | 0.16 | 0.00 |
| 51 | B | 0.21 | 0.04 | 0.05 |
| 52 | B | 0.24 | 0.14 | -0.03 |
| 53 | B | 0.27 | 0.17 | 0.06 |
Call:
lm(formula = s1 ~ Category, data = PCscores)
Residuals:
Min 1Q Median 3Q Max
-0.137063 -0.031294 -0.004309 0.032838 0.119646
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.198269 0.007122 -27.84 <2e-16 ***
CategoryB 0.395197 0.010072 39.24 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.05036 on 98 degrees of freedom
Multiple R-squared: 0.9402, Adjusted R-squared: 0.9395
F-statistic: 1540 on 1 and 98 DF, p-value: < 2.2e-16
\[ f_A(t) = \mu(t) + s_{1, A} \cdot PC1(t)\] \[ f_B(t) = \mu(t) + s_{1, B} \cdot PC1(t)\]
s1 ~ Category explains 94% of the variances2 ~ Category explains 1% of the variance| Category | \(s_1\) |
|---|---|
| A | 0.134 |
| B | -0.134 |
\[ y1_A(t) = \mu_{y1}(t) + \color{red}{s_{1, A}} \cdot PC1_{y1}(t)\] \[ y2_A(t) = \mu_{y2}(t) + \color{red}{s_{1, A}} \cdot PC1_{y2}(t)\]
\[ y1_B(t) = \mu_{y1}(t) + \color{red}{s_{1, B}} \cdot PC1_{y1}(t)\] \[ y2_B(t) = \mu_{y2}(t) + \color{red}{s_{1, B}} \cdot PC1_{y2}(t)\]
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